The Green's function technique is employed to investigate the influence of the boundary condition perturbations in a number of wave propagation problems. The method permits treatment of multiple scattering on random irregularities of a boundary surface which is of particular importance for waveguide applications. For an average Green's function the Dyson type equation has been obtained whose solution represents the coherent part of a point source field in a rough waveguide. The eigenfunction spectrum has also been calculated for such waveguides. By means of mutual wave transformation due to the scattering, the waveguide modes acquire additional (lossless) damping and altered phase velocities. Detailed calculations have been carried through for plane acoustical waveguides with statistically rough walls under the Dirichlet and Neumann conditions. The average field's damping has also been considered for some cases of more complex geometry. In the electromagnetic case the electrical and magnetic solutions are similarly influenced by the wall roughness. Owing to the scattering they acquire longitudinal components of E or H thus becoming guasi-electrical or quasi-magnetic. For these normal waves the damping coefficients (attenuation rates) have been derived. A particular attention is paid to cutoff frequencies in the presence of effective wave conversion to the resonant mode.

Propagation in statistically irregular waveguides--Part I: Average field

A theoretical analysis of incoherent intermode light power diffusion in multimode dielectric waveguides with rough (corrugated) surfaces is presented. The correlation length a of the surface-profile variations is assumed to be sufficiently large (a⪢λ∕2π) to permit light scattering into the outer space only from the modes close to the critical angles of propagation and yet sufficiently small (a⪡d, where d is the average width of the waveguide) to permit direct interaction between a given mode and a large number of neighboring ones. The cases of a one-dimensional (1D) slab waveguide and a two-dimensional cylindrical waveguide (optical fiber) are analyzed, and we find that in both cases the partial differential equations that govern the evolution of the angular light power profile propagating along the waveguide are 1D and of the diffusion type. However, whereas in the former case the effective conductivity coefficient proves to be linearly dependent on the transverse-mode wave number, in the latter one the linear dependence is for the effective diffusion coefficient. The theoretical predictions are in reasonable agreement with experimental results for the intermode power diffusion in multimode (700×700) optical fibers with etched surfaces. The characteristic length of dispersion of a narrow angular power profile evaluated from the correlation length and standard deviation of heights of the surface profile proved to be in good agreement with the experimentally observed changes in the output angular power profiles.

Intermode light diffusion in multimode optical waveguides with rough surfaces.

For over than thirty years it has been believed that the scattering of waves by a randomly rough surface with sufficiently small elevations runs similarly to the scattering by an ideal (unperturbed) surface but having a random impedance. Based on this belief, many of authors applied a habitual simplifying trick from the very outset of handling the wave scattering problem: they replaced the exact boundary condition by its Taylor expansion in terms of the irregularities height keeping only linear term. Following a thorough analysis the authors conclude that a necessary condition for the replacement of an absolutely soft-wall scattering problem with the corresponding impedance approach is the firm certainty that the physical quantities sought are mainly formed by the natural modes whose wavelengths far exceed the average height of the surface elevations.

Is there a physical coherence between the problems of wave scattering from absolutely soft and impedance rough surfaces

Evolution of an emitted signal in a waveguide line is known to be highly sensitive to perturbations of lateral boundaries. Mainly depending on the size of the perturbations, there exist different evolutional scenarios for the signal. Evidently, in extremely imperfect systems where the typical height C of the surface corrugations is comparable with the average channel thickness d, the wave field is expected to be locked by a few adjacent high elevations. In this case a long-distance propagation of the signal through the waveguide is certainly out of the question. However, for telecommunication applications, the most favourable regime is exactly that, when the primary wave packet passes sufficiently long distances from a source owing to multiple re-reflections from the opposite walls. In what follows such a regime will be referred to as a waveguide propagation.

I. M. Fuks

Papers

Average field of a point source in a waveguide with rough walls

Damping of Proper Waves in a Plate with Rough Walls